Testing independence based on Bernstein empirical copula and copula density

Abstract

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Weak convergence of the weighted empirical beta copula process

“…Recently, Belalia et al [1] proposed to use the Bernstein empirical copula [27,33] rather than the empirical copula in the Cramér-von Mises test statistic. Moreover, they constructed new test statistics based on the Bernstein copula density estimator by Bouezmarni et al [5].…”

“…Still, the common value of its coefficients of upper and lower tail dependence is positive and depends on the degrees-of-freedom parameter. In their numerical experiments, Belalia et al [1] found that for such alternatives, the power of the Cramér-von Mises test based on both the empirical copula and the Bernstein empirical copula is particularly weak. Their test statistics based on the Bernstein copula density estimator performed much better.…”

“…The unweighted Cramér-von Mises statistic T n does a poor job in detecting the alternative. The novel statistics δ n and I n in [1] are more powerful, especially the statistic I n , which is a Cramérvon Mises statistic based on the Bernstein copula density estimator. For the weighted Cramér-von Mises statistic T n,γ , the power increases with γ.…”

“…The asymptotic distribution of the statistic under the null hypothesis is an easy corollary of our main result. More interestingly, the inclusion of a weight function leads to a markedly better power against difficult alternatives such as the t copula with zero correlation parameter, with favorable comparisons even to the novel statistics introduced recently by Belalia et al [1]. As a second application we consider the Capéràa-Fougères-Genest estimator [9] of the Pickands dependence function of a multivariate extreme-value copula.…”

Testing the heritability and parent‐of‐origin hypotheses for ages at onset of psoriatic arthritis under biased sampling

“…Most of the references cited above considered only binary status (affected/nonaffected) outcomes and ignored the ages of onset of the disease in their statistical analyses. Testing the heritability and parent-of-origin hypothesis in PsA families consists of assessing the significance of variance component parameters in a mixed effects or a copula model (Zhang & Lin, 2008;Belalia et al, 2017). However, very few of such tests are available for time-toevent outcomes.…”